Calvin Elgot defined iterative theories as algebraic theories in which all guarded recursive equations have unique solutions. One of his major results was a descritpion of a free iterative theory on a signature S as the theory R(S) of all rational S-trees. Later Evelyn Nelson simplified that proof dramaticly by introducing iterative S-algebras and proving that the algebra of all rational S-trees is a free iterative S-algebra. We will describe the Eilenberg-Moore category of all monadic algebras of the monad of free iterative algebras; we call them Elgot algebras. These are S-algebras with solutions of recursive equations satisfying two simple axioms.
One level higher: let us consider the categories Sig of all signatures and Th of all algebraic theories with its canonical forgetful functor U: Th -> Sig. Elgot's result yields the rational-tree monad Rat on the category Sig which is the monad of free iterative theories: to every signature S it assigns therational-trees signature UR(S). What are the monadic algebras of Rat? A surprising answer: precisely the iteration theories of Bloom and Esik. This is a pleasant result in giving the shortest and clearest known definition of iteration theories. (The fundamental monograph of Bloom and Esik conatins a number of different axiomatizations of iteration theories, but none of them is "the main" one, and none of them is easy to remember.)
Traced cocartesian categories were characterizad by Hasegawa as precisely the Conway theories, a natural weakening of iteation theories. The "higher-level game" played above has a variation which yields Conway theories in an analogous way: instead of signatures one starts with all finitary endofunctors of sets, Fin(Set). Every finitary endofunctor H generates a free iteration theory R(H), resulting in a monad R on the category Fin(Set). R has as monadic algebras all Conway theories uniform on base morphisms. This last result is a joint work in progress with Stefan Milius and Jiri Velebil.